Sparse data-driven quadrature rules via ℓ^p-quasi-norm minimization
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We show the use of the focal underdetermined system solver to recover sparse quadrature rules for parametrized integrals from existing data. This algorithm, originally proposed for image and signal reconstruction, relies on an approximated ℓ^p-quasi-norm minimization. The choice of 0<p<1, fits the nature of the constraints to which quadrature rules are subject, thus providing a more suitable formulation for sparse quadrature recovery than the one based on ℓ^1-norm minimization. We also extend an a priori error estimate available for the ℓ^1-norm formulation by considering the error resulting from data compression. Finally we present two numerical examples to illustrate some practical application: the space-time evaluation of the fundamental solution of linear 1D Schrödinger equation where we compare our method with the one based on ℓ^1-norm minimization; the hyper-reduction of a nonlinear diffusion partial differential equation in the framework of reduced basis method.
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